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arxiv: 2505.06195 · v1 · submitted 2025-05-09 · 🧮 math.NA · cs.NA

Stable fully practical finite element methods for axisymmetric Willmore flow

Harald Garcke , Robert N\"urnberg , Quan Zhao This is my paper

Pith reviewed 2026-05-22 15:23 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords axisymmetric Willmore flowfinite element methodsunconditional stabilitygradient flowmean curvatureequidistributionfully discrete schemes
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The pith

Two fully discrete finite element schemes for axisymmetric Willmore flow achieve unconditional stability without remeshing.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops numerical methods to approximate the evolution of closed axisymmetric surfaces under Willmore flow, including cases with spontaneous curvature. It proposes a weak formulation on the generating curve that couples the mean curvature evolution equation with the curvature identity. This combination lets the mean curvature capture the underlying gradient flow while the curvature identity enforces equidistribution of mesh points. A reader would care because the resulting schemes remain stable for any time step size and avoid the need for remeshing, which often destabilizes long-time simulations of curvature-driven surfaces.

Core claim

By expressing the Willmore gradient flow through the mean curvature and treating the in-plane curvature of the generating curve as a Lagrange multiplier for equidistribution, the authors derive two fully discrete finite element schemes whose solutions satisfy an unconditional stability bound. The schemes are formulated directly on the generating curve of a closed axisymmetric surface without boundary, and numerical experiments confirm both convergence and preservation of the equidistribution property.

What carries the argument

The weak formulation that pairs the evolution equation for mean curvature with the curvature identity of the generating curve, allowing the latter to serve as a Lagrange multiplier enforcing equidistribution.

If this is right

  • The discrete solutions satisfy a discrete energy dissipation law for arbitrary time steps.
  • Equidistribution of the mesh points holds automatically at each time level.
  • The schemes remain stable when spontaneous curvature is included.
  • Numerical tests show convergence under successive refinement of the spatial and temporal discretizations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same mean-curvature-plus-Lagrange-multiplier structure might be adapted to other axisymmetric geometric flows such as surface diffusion.
  • Long-time simulations of vesicle or membrane shapes could become more reliable if the equidistribution property persists under more general initial data.
  • Extensions to surfaces with boundary or to non-axisymmetric cases would require new multiplier constructions but could reuse the stability argument.

Load-bearing premise

The surfaces are closed and axisymmetric without boundary, so the problem reduces to a one-dimensional generating curve where the curvature identity can function as a Lagrange multiplier.

What would settle it

A computation in which the discrete energy increases for some positive time step size, or in which mesh points fail to remain equidistributed, would contradict the claimed unconditional stability.

Figures

Figures reproduced from arXiv: 2505.06195 by Harald Garcke, Quan Zhao, Robert N\"urnberg.

Figure 1
Figure 1. Figure 1: Sketch of Γ and S, as well as the unit vectors ~e1, ~e2 and ~e3. We consider the axisymmetric case and assume that the surface S(t) satisfies the rotational symmetry with respect to the x2-axis, as shown in [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: [κ = 0, J = 128, ∆t = 10−3 ] Evolution of an initial disk of dimension of 7× 1× 7. We plot Γm at times t = 0, 0.5, 1, · · · , 3, 10 and visualize the axisymmetric surfaces at t = 0.5 and t = 10. Example 3: We conduct an experiment for a rounded cylinder of total dimension 2 × 6 × 2 in the case of κ = −2, which was also considered in [6, Fig.9]. The numerical results are shown in [PITH_FULL_IMAGE:figures/f… view at source ↗
Figure 3
Figure 3. Figure 3: Plots of the discrete energy and the mesh ratio R [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: [κ = −1.25, J = 128, ∆t = 10−3 ] Evolution of an initial disk of dimension of 7×1×7. We plot Γm at times t = 0, 0.5, 1, · · · , 3, 10 and visualize the axisymmetric surfaces at t = 0.5 and t = 10. 0 2 4 6 8 10 t 0 20 40 60 Energy 0 2 4 6 8 10 t 1 1.2 1.4 1.6 1.8 2 [PITH_FULL_IMAGE:figures/full_fig_p020_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Plots of the discrete energy and the mesh ratio R [PITH_FULL_IMAGE:figures/full_fig_p020_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: [κ = −2, J = 128, ∆t = 2.5 × 10−4 ] Evolution of an initial rounded cylinder of dimension 2 × 6 × 2. We plot Γm at times t = 0, 0.1, 0.2, · · · , 1 and visualize the axisymmetric surface generated by Γm at time t = 1. On the right are plots of the discrete energy and the mesh ratio Rm [PITH_FULL_IMAGE:figures/full_fig_p021_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: [κ = 0, J = 128, ∆t = 2.5 × 10−4 ] Evolution of an initial flat annulus shape towards a Clifford torus. We plot Γm at times t = 0, 0.5, · · · , 5, 20 and visualize the axisymmetric surfaces generated by Γm at t = 0.5 and t = 20. On the bottom are plots of the discrete energy and the mesh ratio Rm. Example 5: We consider the evolution of a genus-1 surface, where the generating curve Γ(0) is given by an elon… view at source ↗
Figure 8
Figure 8. Figure 8: [κ = −2, J = 256, ∆t = 6.25×10−5 ] Evolution of an initial flat annulus shape towards a thin torus. We plot Γm at times t = 0, 0.1, · · · , 1, 2 and visualize the axisymmetric surface generated by Γm at t = 2. On the bottom are plots of the discrete energy and the mesh ratio Rm. consider the case of κ = 0 and the numerical results are shown in [PITH_FULL_IMAGE:figures/full_fig_p023_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: [κ = 1, J = 256, ∆t = 6.25 × 10−5 ] Evolution of an initial flat annulus shape. We plot Γ m at times t = 0, 0.5, · · · , 2.5, 2.6, 2.7, 2.8, 2.9, 3, 4, where the plots for t = 3 and t = 4 lie on top of each other. We also visualize the axisymmetric surface generated by Γm at t = 4. On the bottom are plots of the discrete energy, the mesh ratio Rm and the distance to the rotational axis. Example 6: In our l… view at source ↗
Figure 10
Figure 10. Figure 10: [κ = 2, J = 128, ∆t = 2.5 × 10−4 ] Evolution of an initial flat annulus shape until its pinch-off. We plot Γm at times t = 0, 0.1, · · · , 0.7, 0.72, 0.75, 0.77 and visualize the axisymmetric surface generated by Γm at t = 0.7. On the bottom are plots of the discrete energy, the mesh ratio Rm and the distance to the rotational axis. shown to satisfy unconditional stability estimates, as well as an asympto… view at source ↗
read the original abstract

We consider fully discrete numerical approximations for axisymmetric Willmore flow that are unconditionally stable and work reliably without remeshing. We restrict our attention to surfaces without boundary, but allow for spontaneous curvature effects. The axisymmetric setting allows us to formulate our schemes in terms of the generating curve of the considered surface. We propose a novel weak formulation, that combines an evolution equation for the surface's mean curvature and the curvature identity of the generating curve. The mean curvature is used to describe the gradient flow structure, which enables an unconditional stability result for the discrete solutions. The generating curve's curvature, on the other hand, describes the surface's in-plane principal curvature and plays the role of a Lagrange multiplier for an equidistribution property on the discrete level. We introduce two fully discrete schemes and prove their unconditional stability. Numerical results are provided to confirm the convergence, stability and equidistribution properties of the introduced schemes.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper develops two fully discrete finite element schemes for axisymmetric Willmore flow on closed surfaces without boundary, allowing spontaneous curvature. It introduces a mixed weak formulation that couples an evolution equation for mean curvature (capturing the L2-gradient flow structure of the Willmore energy) with the curvature identity of the generating curve (enforced weakly to enforce discrete equidistribution). Unconditional stability is proved for both schemes, and numerical experiments confirm convergence, stability, and equidistribution properties.

Significance. If the stability analysis holds, the work delivers practical, remeshing-free methods with no time-step restrictions for a geometrically nonlinear fourth-order flow in a reduced-dimensional setting. The use of the curvature identity as a Lagrange multiplier for exact discrete equidistribution is a notable technical contribution that strengthens the practical utility of the schemes.

minor comments (2)
  1. [Section 3] In the description of the weak formulation, the precise incorporation of spontaneous curvature into the energy and the resulting variational terms could be stated more explicitly to aid readers unfamiliar with the axisymmetric reduction.
  2. [Section 6] The numerical section would benefit from a brief discussion of how the weighted Sobolev spaces are discretized in practice (e.g., quadrature rules for the r-weighted integrals) to ensure reproducibility of the reported convergence rates.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our manuscript on unconditionally stable fully discrete finite element schemes for axisymmetric Willmore flow and for recommending minor revision. We appreciate the recognition of the practical utility of the schemes and the technical contribution of the curvature identity as a Lagrange multiplier for equidistribution. Since no specific major comments were raised, we will focus on minor improvements to presentation and clarity in the revised version.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper's central result is unconditional stability of two fully discrete finite element schemes for axisymmetric Willmore flow. This follows directly from a mixed weak formulation that encodes the L2-gradient flow structure via an evolution equation for mean curvature, combined with the curvature identity of the generating curve enforced weakly. The discrete energy estimate is obtained by testing the scheme with the discrete velocity and curvature variables, producing exact cancellation independent of time-step size. This is a standard energy-dissipation argument in the numerical analysis of gradient flows and does not reduce to any fitted parameter, self-definition, or self-citation chain. The axisymmetric reduction to a generating curve for closed surfaces without boundary is a standard geometric simplification with no hidden assumptions that presuppose the stability result.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work relies on standard assumptions from finite element theory and geometric analysis for weak formulations of curvature flows; no ad-hoc free parameters or invented entities are introduced beyond the discretization itself.

axioms (2)
  • standard math Standard Sobolev space setting and weak formulation for mean curvature flow on axisymmetric surfaces
    Invoked to derive the evolution equation and stability from the gradient flow structure.
  • domain assumption Curvature identity of the generating curve holds in the discrete setting to enforce equidistribution
    Used as Lagrange multiplier; central to the equidistribution property claimed.

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